SSD-offloaded training offers a practical and promising approach to making LLM training cost-effective. Building on gradient accumulation with micro-batches, this paper introduces GreedySnake, a new SSD-offloaded training system that employs vertical scheduling, which executes all microbatches of a layer before proceeding to the next. Compared to existing systems that use horizontal scheduling (i.e., executing micro-batches sequentially), GreedySnake achieves higher training throughput with smaller batch sizes, bringing the system much closer to the ideal scenario predicted by the roofline model. To further mitigate the I/O bottleneck, GreedySnake overlaps part of the optimization step with the forward pass of the next iteration. Experimental results on A100 GPUs show that GreedySnake achieves saturated training throughput improvements over ZeRO-Infinity: 1.96x on 1 GPU and 1.93x on 4 GPUs for GPT-65B, and 2.53x on 1 GPU for GPT-175B.